Simplify and expand the following expression: $ \dfrac{4}{z + 7}+ \dfrac{4}{z + 1}+ \dfrac{2z}{z^2 + 8z + 7} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{2z}{z^2 + 8z + 7} = \dfrac{2z}{(z + 7)(z + 1)}$ Now we have: $ \dfrac{4}{z + 7}+ \dfrac{4}{z + 1}+ \dfrac{2z}{(z + 7)(z + 1)} $ The least common multiple of the denominators is: $ (z + 7)(z + 1)$ In order to get the first term over $(z + 7)(z + 1)$ , multiply by $\dfrac{z + 1}{z + 1}$ $ \dfrac{4}{z + 7} \times \dfrac{z + 1}{z + 1} = \dfrac{4(z + 1)}{(z + 7)(z + 1)} $ In order to get the second term over $(z + 7)(z + 1)$ , multiply by $\dfrac{z + 7}{z + 7}$ $ \dfrac{4}{z + 1} \times \dfrac{z + 7}{z + 7} = \dfrac{4(z + 7)}{(z + 7)(z + 1)} $ Now we have: $ \dfrac{4(z + 1)}{(z + 7)(z + 1)} + \dfrac{4(z + 7)}{(z + 7)(z + 1)} + \dfrac{2z}{(z + 7)(z + 1)} $ $ = \dfrac{ 4(z + 1) + 4(z + 7) + 2z} {(z + 7)(z + 1)} $ Expand: $ = \dfrac{4z + 4 + 4z + 28 + 2z}{z^2 + 8z + 7} $ $ = \dfrac{10z + 32}{z^2 + 8z + 7}$